Lord Kelvin: An account of his scientific life and work

by Andrew Gray

Fig. 6.

The inverse of a circle is another circle, and therefore the inverse of a sphere is another sphere, and the inverse of a straight line is a circle passing through the centre of inversion, and of an infinite plane a sphere passing through the centre of inversion. Obviously the inverse of a sphere concentric with the sphere of inversion is a concentric sphere.

The line P'Q' is of course not the inverse of the line PQ, which has for its inverse the circle passing through the three points O, P', Q', as indicated in Fig. [6].

The following results are easily proved.

A locus and its inverse cut any line OP at the same angle.

To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on one side of the surface of the sphere of inversion there is a system of charges aq₁ ⁄ f₁, aq₂ ⁄ f₂, ... on the other side of the spherical surface [OP₁ = f₁, OP₂ = f₂]. This inverse system, as we shall call it, produces the same potential at any point of the sphere of inversion, as does the direct system from which it is derived.

If V, V' be the potentials produced by the whole direct system at Q, and by the whole inverse system at Q', V' ⁄ V = r ⁄ a = a ⁄ r', where OQ = r, OQ' = r'.

Thus if V is constant over any surface S', V' is not a constant over the inverse surface S', unless r is a constant, that is, unless the surface S' is a sphere concentric with the sphere of inversion, in which case the inverse surface is concentric with it and is an equipotential surface of the inverse distribution.

Further, if q be distributed over an element dS of a surface, the inverse charge aq ⁄ f will be distributed over the corresponding element dS' of the inverse surface. But dS' ⁄ dS = a⁴ ⁄ f⁴ = f'⁴ ⁄ a⁴ where f, f' are the distances of O from dS and dS'. Thus if s be the density on dS and s' the inverse density on dS' we have s' ⁄ s = a³ ⁄ f'³ = f³ ⁄ a³.

When V is constant over the direct surface, while r has different values for different directions of OQ, the different points of the inverse surface may be brought to zero potential by placing at O a charge − aV. For this will produce at Q' a potential − aV ⁄ r' which with V' will give at Q' a potential zero. This shows that V' is the potential of the induced distribution on S' due to a charge − aV at O, or that − V' is the potential due to the induced charge on S' produced by the charge aV at O.